A uniform magnetic field of magnitude along is described

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Magnetostatics

Invariance, Ch Integrating forms: Stokes's theorem, holes and Maxwell's equations

Problem

A uniform magnetic field of magnitude B0=0.40 TB_0=0.40\,\mathrm T along zz is described, on the xyxy-plane, by the two-form F=B0 dx∧dyF=B_0\,\mathrm dx\wedge\mathrm dy. Two candidate vector potentials are proposed: the symmetric-gauge form A1=12B0(x dy−y dx)A_1=\tfrac12B_0(x\,\mathrm dy-y\,\mathrm dx) and the Landau-gauge form A2=B0x dyA_2=B_0x\,\mathrm dy. ((a)) Verify that dA1=dA2=F\mathrm dA_1=\mathrm dA_2=F. ((b)) Show that A1−A2=dλA_1-A_2=\mathrm d\lambda for some function λ(x,y)\lambda(x,y), find λ\lambda, and evaluate it at the point (x,y)=(2.0 m,3.0 m)(x,y)=(2.0\,\mathrm m,3.0\,\mathrm m). ((c)) Writing A2=B0x dyA_2=B_0x\,\mathrm dy in the coordinate basis {dx,dy}\{\mathrm dx,\mathrm dy\} (in Cartesian axes these have the same components as the dual basis vectors e1,e2\mathbf e^1,\mathbf e^2 of the general-bases chapter), read off its covariant components (A2)1,(A2)2(A_2)_1,(A_2)_2, then raise the index with the Euclidean metric gij=δijg_{ij}=\delta_{ij} to recover the familiar vector potential A=(A21,A22,0)\mathbf A=(A_2^1,A_2^2,0) of a uniform field, and identify it with the Landau-gauge potential held from an earlier electromagnetism or solid-state course.

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